Statistical and Probabilistic models

Statistical and Probabilistic models are commonly used to predict service lifetime of assets and likelihood of occurrence of an event. Data about condition and features of pipes are necessary to generate expected outcome through statistical analysis. Therefore, large and long term historical databases are required.

Kleiner and Rajani (2001) made a comprehensive review of statistical and probabilistic models about structural deterioration of water mains until 2000. They divided the models into four categories based on the approach of problem solving. It includes time-exponential models (Shamir and Howard, 1979; Walski and Pelliccia, 1982; Clark et al., 1982), time- linear models (McMullen, 1982; Kettler and Goulter, 1985; Jacobs and Karney, 1994), Probabilistic multi-variate models, proportional hazards and accelerated life (Marks et al., 1985; Andreou et al., 1987; Marks et al., 1987; Bremond, 1997; Constantine and Darroch, 1993; Miller, 1993; Constantine et al., 1996; Lei, 1997; Eisenbeis et al., 1999) and

Probabilistic single-variate group models (Kulkarni et al., 1986; Goulter et al., 1993; Deb et al., 1996; Mavin, 1996; Herz, 1998; Gustafson and Clancy, 1999).

Le Gat and Eisenbeis (2000) used maintenance records to forecast failures in water networks in different materials. They calculated the times and number of failure using Weibull Proportional Hazard Model (WPHM). The survival function considered the factors affecting the failure of the pipe. The model used two databases to check both observed and predicted failures. It underestimated one of the case studies because of the increased pipe deterioration and lack of data. Moreover, there is no evidence of model validation in this study. Factors considered in this study are Pipe components (length, age, diameter, pressure and material), soil type, traffic load, and supply method such as gravity and pumping. History of previous failures was taken into account for calculation.

Park and Loganathan (2002) proposed a methodology for efficient optimal replacement of pipes in water distribution systems. The method uses threshold break rates and failure prediction models to determine the time. They considered the optimal threshold break rate as a function of pipe diameter and costs of replacement or repair. By setting the threshold break rate equal to estimated rate by failure prediction model, the replacement time is calculated. The inputs of the model are number of breaks per 1000 ft of pipe, growth rate coefficient, annual interest rate, repair, and replacement costs for the pipe length. In addition, Loganathan et al. (2002) proposed a threshold break rate for pipeline replacement in water distribution systems.

Pelletier et al. (2003) Modeled water pipe break rates based on breakage history and pipe components (diameter, length, material and age). The authors estimated present and future structural states of water main by use of pipe break model. Statistical functions of survival,

probability distribution and hazard were utilized to represent the Weibull and exponential distributions. Three case studies were performed to analyze the model. The survival functions were versus time and they are based on the Weibull distribution of the pipe failure through pipe age. Results show that material and installation method affect pipe deterioration. This model does not take into account the corrosion and leakage. It only considers breaks due to natural aging of pipes. Other inputs of the model are soil type and land use.

DeSilva et al. (2006) proposed a condition assessment and probabilistic analysis to estimate failure rates in buried metallic pipeline. Most of the water networks confront the problem of data scarcity since there is no complete database about breaks history of water mains. Therefore, condition assessment is used to find the deterioration rates and probabilities of failure in a distribution system. First, a Weibull probability distribution function is used to estimate maximum corrosion rate. Then, the distribution is extrapolated over larger target area. After that, it is converted to corresponding normal distribution functions. Later, the probability of failure was determined using first-order-second-moment analysis. In validation, results of binomial probability process show the relationship between failure rate and time. Overall, the model entails prediction of failure of entire pipeline assessing condition of selected sections. The model takes into account variables of maximum applied stress, critical stress required for failure related to external and internal galvanic corrosion, internal pressure, pipe wall thickness and radius of the pipe.

Vanrenterghem-Raven (2007) proposed a proportional hazard model (PHM) to measure risk factors of structural degradation and break rates of water distribution system through running PHM while one variable at a time is considered. PHM is a statistical method

applicable for renewable processes. This methodology calculates expected breaks per pipe using hazard rate. Inputs required for this model is pipe components (length, material (steel/non-steel), diameter and age) and environmental factors (traffic, location, water zones and highways).

Davis et al. (2007) proposed a physical probabilistic model to predict failure rates in buried PVC pipeline. The model used internal defects resulting from internal pressure to determine failure rate. The output of the model was compared to the observed data form different municipalities in United Kingdom. Results show that predicted curves are in a favorable agreement with observed data. A Monte Carlo simulation estimates the lifetime probability distribution. Variables used in Monte Carlo simulation are number of segments, length, incremental time period, material short-term properties (fracture toughness, yield strength and Young’s modulus), slow crack growth parameters, visco-elastic parameter for reduction in Young’s modulus, outer and inner pipe diameter, maximum internal pressure in each segment, soil properties (cover depth, unit weight and modulus), surface load and residual hoop stress.

Poulton et al. (2009) measured the impact of pipe length on break predictions in water mains. They utilized Linearly Extended Yule Process (LEYP) to find break predictions for each segment. Calculations were done based on intensity function and by aid of LEYP. Intensity function depends on age in the form of Weibull model, number of previous events in the form of LEYP and vector of covariates in the form of Cox proportional hazard model. After model verification, results show that model is not sensitive to small segments, which means that pipe length does not affect breaks since it is merely related to pipe age, water

pressure and soil type. Input parameters are pipe diameter, length, installation year, soil type, traffic level, water pressure, type of accident, and date of intervention.

Dehghan et al. (2008) presented a parametric model using probabilistic analysis of structural failure of water pipes. Due to the fact that the theoretical failure rates do not depend on time, the authors tested the steadiness of the failure rates. Results of parametric models are valid when failure rate is considered as a stationary random process. Verification of the model through a case study showed that variables should be updated to represent time dependent nature of failure process. The input parameters are pipe material, diameter and location. Developing a nonparametric model was proposed to investigate the time dependent failure process correctly.

Dehghan et al. (2008) also proposed a nonparametric methodology for Probabilistic failure prediction for pipeline deterioration. The nonparametric methodology considers those factors that are not constant in reality. In this method, probability of failure was estimated by maximum likelihood in order to evaluate number of failure per time and confidence intervals. The methodology was tested through the observed data of water network in Australia. Results show a good agreement between observed and predicted data. As the model doesn’t intend to predict single component failure, mains are analyzed in groups taking into account their material type, diameter, and location.

Davis and Marlow (2008) proposed a physical probabilistic failure model utilizing Weibull probability distribution for quantifying economic lifetime for asset management of large diameter cast iron pipelines. Since enough data about failure was not available, condition assessment was employed to determine remaining life time. To determine the corrosion rate, the model investigated selected parts of the pipe. This model evaluates failure time

and economic life time. It only shows longitudinal fracture because it solely calculates internal pressure and in-plane bending. Inputs are pipe diameter, age, thickness, external loads from soil and surface loads, maximum corrosion rate and tensile strength.

Kleiner and Rajani (2008) checked prioritizing individual water mains for renewal using non-homogenous Poisson model. They divided the input parameters of pipe material, diameter, length, installation year, climate, X-Y coordinates of pipe nodes, break date, and type into 3 classes of pipe dependent, time dependent and pipe-time dependent. The model was first trained using maximum likelihood method with a Lipschitz Global Optimizer (LGO) algorithm. Then it was validated by forecasting the number of breaks in a validation period. Afterward, the observed and predicted failures are compared. Results show that model is appropriate and is able to analyze the covariates at group and pipe level.

Davis et al. (2008) presented failure prediction and optimal scheduling of replacement in asbestos cement (AC) water pipes through probabilistic failure model. This model utilizes residual strength to find the deterioration rate employing Weibull distribution. However since there were some differences between data produced and observed, Hertz distribution was employed to model the uncertainty. Results of the verification show that predicted life time using Hertz distribution are similar to empirical lifetime evaluated by Monte Carlo simulation. The data requirements in this article are pipe diameter, thickness, depth, age, internal pressure, unit weight of the surrounding soil, dynamic traffic load, and deterioration rate.

Berardi et al. (2008) suggested using EPR in developing deterioration models for water distribution systems. EPR is divided into two steps: searching for the best model structures using GA and parameter estimation for an assumed structure using least squares method.

EPR performs a multi-objective search to find the best model. Savic (2009) also used same methodology and case study in his article “The Use of Data Driven Methodologies for Prediction of Water and Wastewater Asset Failure”. Since it is not possible to use EPR for direct calculation of failure rate, two different processes were used to find a pipe condition. The processes were developing general failure model and implementing multi-objective approach for pipe rehabilitation planning. Data requirements for model development are pipe parameters of age, diameter, length and number of properties.

Moglia et al. (2008) checked the strong exploration of a cast iron pipe failure model. The first proposed model was improved through several assumptions i.e. time dependent corrosion rates; stochastic pipe wall thickness; stochastic loads; and lower limit on the tensile strength. The new model determines maximum corrosion rates through evaluating the nominal tensile strength. In this model, only failures caused by corrosion or fractures are considered. Since the model has numerous assumptions, it is not generalized. After validation, the predicted and observed data were similar. The input parameters for this model are pipe components (thickness, age, failure and installation year, diameter and length), internal pressure, external and soil load, failure exposure, corrosion rate, number of observed failures and the tensile strength of the pipe samples.

Wang et al. (2009) developed deterioration model to predict annual break rates of water mains respect to pipe material. The method consisted of finding the best subset regression to determine the best relationship between failure rate and variables such as pipe age, diameter and length. In addition to variables mentioned, break records, pipe depth and material are considered. The model was verified using a case study in Canada. Sensitivity analysis was performed to find the ways that different variables affect the annual breaks.

The regression models were verified by F-test and t-test. The model doesn’t consider repair history, cathodic protection and soil condition and is not able to predict the next failure. Wood and Lence (2009) used water main break data to improve asset management for small and medium utilities. Time-linear and time-exponential equations were developed. Results show that with the exception of cast iron (CI), predictions in asbestos cement (AC) and ductile iron (DI) are more accurate utilizing the time-linear equation. Required input data includes break history, ground surface material, pipe material, diameter and age. Wang et al. (2010) proposed an assessment model of water pipe condition using Bayesian inference. In this model, the relative effects of each factors on model performance was evaluated and those with smallest effect was excluded. Comparing the model output and observed data, pipe age and diameter proved to have the most effect on pipe condition. Verification of model shows that proposed model is within good compliance. Input variables are pipe components (diameter, age, material, depth, inner and outer coating), pressure head, number of road lanes, electric recharge, bedding and soil conditions. As can be seen from statistical and probabilistic models, statistical models needs large number of long term observed field data while probabilistic models are convenient for databases that have little information. In most of the probabilistic models, condition assessment of a pipe is only analyzed for a pipe section not the entire pipeline. These models only predict the failure of a water pipe and there is no generalization of them.